_divideComplex, real part

Percentage Accurate: 61.1% → 88.4%

Time: 13.2s

Alternatives: 13

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.7e+147)
   (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
   (if (<= y.im 7.5e+110)
     (*
      (/ 1.0 (hypot y.re y.im))
      (* y.re (/ (fma x.im (/ y.im y.re) x.re) (hypot y.re y.im))))
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.7e+147) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 7.5e+110) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (y_46_re * (fma(x_46_im, (y_46_im / y_46_re), x_46_re) / hypot(y_46_re, y_46_im)));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.7e+147)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 7.5e+110)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(y_46_re * Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / hypot(y_46_re, y_46_im))));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.7e+147], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+110], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re * N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.69999999999999991e147

   1. Initial program 25.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0 25.1%

      \[\leadsto \frac{\color{blue}{x.im \cdot y.im}}{y.re \cdot y.re + y.im \cdot y.im} \]
   4. Step-by-step derivation

      1. *-commutative 25.1%

         \[\leadsto \frac{\color{blue}{y.im \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt 25.1%

         \[\leadsto \frac{y.im \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. hypot-undefine 25.1%

         \[\leadsto \frac{y.im \cdot x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      4. hypot-undefine 25.1%

         \[\leadsto \frac{y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      5. times-frac 97.4%

         \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -5.69999999999999991e147 < y.im < 7.5e110

   1. Initial program 67.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 64.7%

      \[\leadsto \frac{\color{blue}{y.re \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
   4. Step-by-step derivation

      1. associate-/l* 64.2%

         \[\leadsto \frac{y.re \cdot \left(x.re + \color{blue}{x.im \cdot \frac{y.im}{y.re}}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   5. Simplified 64.2%

      \[\leadsto \frac{\color{blue}{y.re \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 64.2%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(y.re \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt 64.2%

         \[\leadsto \frac{1 \cdot \left(y.re \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. hypot-undefine 64.2%

         \[\leadsto \frac{1 \cdot \left(y.re \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      4. hypot-undefine 64.2%

         \[\leadsto \frac{1 \cdot \left(y.re \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      5. times-frac 78.2%

         \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      6. +-commutative 78.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot \color{blue}{\left(x.im \cdot \frac{y.im}{y.re} + x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      7. fma-define 78.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot \color{blue}{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   7. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot \mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
   8. Step-by-step derivation

      1. associate-/l* 95.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(y.re \cdot \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]

   9. Simplified 95.0%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]

   if 7.5e110 < y.im

   1. Initial program 45.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 80.0%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. associate-/l* 91.5%

         \[\leadsto \frac{x.im + \color{blue}{x.re \cdot \frac{y.re}{y.im}}}{y.im} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))
      INFINITY)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))
   (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 73.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 73.0%

         \[\leadsto \color{blue}{1 \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

      2. associate-*r/ 73.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{y.re \cdot y.re + y.im \cdot y.im}} \]

      3. fma-define 73.0%

         \[\leadsto \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      4. add-sqr-sqrt 73.0%

         \[\leadsto \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      5. times-frac 73.0%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      6. fma-define 73.0%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      7. hypot-define 73.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      8. fma-define 73.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      9. fma-define 73.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \]

      10. hypot-define 94.4%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 94.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 0.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0 1.9%

      \[\leadsto \frac{\color{blue}{x.im \cdot y.im}}{y.re \cdot y.re + y.im \cdot y.im} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \frac{\color{blue}{y.im \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt 1.9%

         \[\leadsto \frac{y.im \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. hypot-undefine 1.9%

         \[\leadsto \frac{y.im \cdot x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      4. hypot-undefine 1.9%

         \[\leadsto \frac{y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      5. times-frac 53.9%

         \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.2e+114)
   (/ (+ x.im (* y.re (* x.re (/ 1.0 y.im)))) y.im)
   (if (<= y.im -3.8e-76)
     (/ (+ (* y.re x.re) (* y.im x.im)) (fma y.re y.re (* y.im y.im)))
     (if (<= y.im 2.8e-20)
       (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
       (if (<= y.im 2e+113)
         (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
         (/ (+ x.im (* x.re (/ y.re y.im))) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -7.2e+114) {
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	} else if (y_46_im <= -3.8e-76) {
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 2.8e-20) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2e+113) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.2e+114)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re * Float64(1.0 / y_46_im)))) / y_46_im);
	elseif (y_46_im <= -3.8e-76)
		tmp = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2.8e-20)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 2e+113)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -7.2e+114], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.8e-76], N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.8e-20], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2e+113], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -7.2000000000000001e114

   1. Initial program 32.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 89.2%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. div-inv 89.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(x.re \cdot y.re\right) \cdot \frac{1}{y.im}}}{y.im} \]

      2. *-commutative 89.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(y.re \cdot x.re\right)} \cdot \frac{1}{y.im}}{y.im} \]

      3. associate-*l* 93.4%

         \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   5. Applied egg-rr 93.4%

      \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   if -7.2000000000000001e114 < y.im < -3.8000000000000002e-76

   1. Initial program 85.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. fma-define 85.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. fma-define 85.0%

         \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 85.0%

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

   6. Applied egg-rr 85.0%

      \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

   if -3.8000000000000002e-76 < y.im < 2.8000000000000003e-20

   1. Initial program 61.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 85.2%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. associate-/l* 85.4%

         \[\leadsto \frac{x.re + \color{blue}{x.im \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}} \]

   if 2.8000000000000003e-20 < y.im < 2e113

   1. Initial program 74.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0 62.8%

      \[\leadsto \frac{\color{blue}{x.im \cdot y.im}}{y.re \cdot y.re + y.im \cdot y.im} \]
   4. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{\color{blue}{y.im \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt 62.8%

         \[\leadsto \frac{y.im \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. hypot-undefine 62.8%

         \[\leadsto \frac{y.im \cdot x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      4. hypot-undefine 62.8%

         \[\leadsto \frac{y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      5. times-frac 88.1%

         \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if 2e113 < y.im

   1. Initial program 45.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 80.0%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. associate-/l* 91.5%

         \[\leadsto \frac{x.im + \color{blue}{x.re \cdot \frac{y.re}{y.im}}}{y.im} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.re + y.im \cdot x.im\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.9 \cdot 10^{-118}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re x.re) (* y.im x.im))))
   (if (<= y.im -6.5e+114)
     (/ (+ x.im (* y.re (* x.re (/ 1.0 y.im)))) y.im)
     (if (<= y.im -4.5e-78)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 6.9e-118)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 4.3e+83)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_re) + (y_46_im * x_46_im);
	double tmp;
	if (y_46_im <= -6.5e+114) {
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	} else if (y_46_im <= -4.5e-78) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 6.9e-118) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 4.3e+83) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_im <= -6.5e+114)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re * Float64(1.0 / y_46_im)))) / y_46_im);
	elseif (y_46_im <= -4.5e-78)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 6.9e-118)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 4.3e+83)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.5e+114], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.5e-78], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.9e-118], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.3e+83], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -6.5000000000000001e114

   1. Initial program 32.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 89.2%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. div-inv 89.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(x.re \cdot y.re\right) \cdot \frac{1}{y.im}}}{y.im} \]

      2. *-commutative 89.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(y.re \cdot x.re\right)} \cdot \frac{1}{y.im}}{y.im} \]

      3. associate-*l* 93.4%

         \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   5. Applied egg-rr 93.4%

      \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   if -6.5000000000000001e114 < y.im < -4.5e-78

   1. Initial program 85.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. fma-define 85.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. fma-define 85.0%

         \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 85.0%

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

   6. Applied egg-rr 85.0%

      \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

   if -4.5e-78 < y.im < 6.9000000000000002e-118

   1. Initial program 58.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 88.8%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. associate-/l* 89.1%

         \[\leadsto \frac{x.re + \color{blue}{x.im \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}} \]

   if 6.9000000000000002e-118 < y.im < 4.3e83

   1. Initial program 80.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   if 4.3e83 < y.im

   1. Initial program 47.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 76.3%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. associate-/l* 85.9%

         \[\leadsto \frac{x.im + \color{blue}{x.re \cdot \frac{y.re}{y.im}}}{y.im} \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.9 \cdot 10^{-118}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -6e+114)
     (/ (+ x.im (* y.re (* x.re (/ 1.0 y.im)))) y.im)
     (if (<= y.im -9.2e-79)
       t_0
       (if (<= y.im 1.1e-117)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 3.1e+71)
           t_0
           (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -6e+114) {
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	} else if (y_46_im <= -9.2e-79) {
		tmp = t_0;
	} else if (y_46_im <= 1.1e-117) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.1e+71) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-6d+114)) then
        tmp = (x_46im + (y_46re * (x_46re * (1.0d0 / y_46im)))) / y_46im
    else if (y_46im <= (-9.2d-79)) then
        tmp = t_0
    else if (y_46im <= 1.1d-117) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 3.1d+71) then
        tmp = t_0
    else
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -6e+114) {
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	} else if (y_46_im <= -9.2e-79) {
		tmp = t_0;
	} else if (y_46_im <= 1.1e-117) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.1e+71) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -6e+114:
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im
	elif y_46_im <= -9.2e-79:
		tmp = t_0
	elif y_46_im <= 1.1e-117:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 3.1e+71:
		tmp = t_0
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -6e+114)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re * Float64(1.0 / y_46_im)))) / y_46_im);
	elseif (y_46_im <= -9.2e-79)
		tmp = t_0;
	elseif (y_46_im <= 1.1e-117)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 3.1e+71)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -6e+114)
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	elseif (y_46_im <= -9.2e-79)
		tmp = t_0;
	elseif (y_46_im <= 1.1e-117)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 3.1e+71)
		tmp = t_0;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+114], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9.2e-79], t$95$0, If[LessEqual[y$46$im, 1.1e-117], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+71], t$95$0, N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.0000000000000001e114

   1. Initial program 32.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 89.2%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. div-inv 89.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(x.re \cdot y.re\right) \cdot \frac{1}{y.im}}}{y.im} \]

      2. *-commutative 89.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(y.re \cdot x.re\right)} \cdot \frac{1}{y.im}}{y.im} \]

      3. associate-*l* 93.4%

         \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   5. Applied egg-rr 93.4%

      \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   if -6.0000000000000001e114 < y.im < -9.20000000000000047e-79 or 1.1000000000000001e-117 < y.im < 3.10000000000000018e71

   1. Initial program 83.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   if -9.20000000000000047e-79 < y.im < 1.1000000000000001e-117

   1. Initial program 58.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 88.8%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. associate-/l* 89.1%

         \[\leadsto \frac{x.re + \color{blue}{x.im \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}} \]

   if 3.10000000000000018e71 < y.im

   1. Initial program 47.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 76.3%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. associate-/l* 85.9%

         \[\leadsto \frac{x.im + \color{blue}{x.re \cdot \frac{y.re}{y.im}}}{y.im} \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.65 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3600000000000:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-54} \lor \neg \left(y.im \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.65e+147)
   (/ x.im y.im)
   (if (<= y.im -3600000000000.0)
     (* (/ y.re y.im) (/ x.re y.im))
     (if (or (<= y.im -6.2e-54) (not (<= y.im 7.2e-9)))
       (/ x.im y.im)
       (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.65e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -3600000000000.0) {
		tmp = (y_46_re / y_46_im) * (x_46_re / y_46_im);
	} else if ((y_46_im <= -6.2e-54) || !(y_46_im <= 7.2e-9)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.65d+147)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-3600000000000.0d0)) then
        tmp = (y_46re / y_46im) * (x_46re / y_46im)
    else if ((y_46im <= (-6.2d-54)) .or. (.not. (y_46im <= 7.2d-9))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.65e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -3600000000000.0) {
		tmp = (y_46_re / y_46_im) * (x_46_re / y_46_im);
	} else if ((y_46_im <= -6.2e-54) || !(y_46_im <= 7.2e-9)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.65e+147:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -3600000000000.0:
		tmp = (y_46_re / y_46_im) * (x_46_re / y_46_im)
	elif (y_46_im <= -6.2e-54) or not (y_46_im <= 7.2e-9):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.65e+147)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -3600000000000.0)
		tmp = Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im));
	elseif ((y_46_im <= -6.2e-54) || !(y_46_im <= 7.2e-9))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.65e+147)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -3600000000000.0)
		tmp = (y_46_re / y_46_im) * (x_46_re / y_46_im);
	elseif ((y_46_im <= -6.2e-54) || ~((y_46_im <= 7.2e-9)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.65e+147], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3600000000000.0], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$im, -6.2e-54], N[Not[LessEqual[y$46$im, 7.2e-9]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.6500000000000001e147 or -3.6e12 < y.im < -6.20000000000000008e-54 or 7.2e-9 < y.im

   1. Initial program 48.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 71.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -2.6500000000000001e147 < y.im < -3.6e12

   1. Initial program 72.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 58.4%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   4. Taylor expanded in x.im around 0 41.2%

      \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im}}}{y.im} \]
   5. Step-by-step derivation

      1. *-commutative 41.2%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im}}{y.im} \]

      2. associate-*r/ 47.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]

   6. Simplified 47.4%

      \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]
   7. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \frac{\color{blue}{\frac{x.re}{y.im} \cdot y.re}}{y.im} \]

      2. associate-/l* 47.4%

         \[\leadsto \color{blue}{\frac{x.re}{y.im} \cdot \frac{y.re}{y.im}} \]

   8. Applied egg-rr 47.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.im} \cdot \frac{y.re}{y.im}} \]

   if -6.20000000000000008e-54 < y.im < 7.2e-9

   1. Initial program 63.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 68.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.65 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3600000000000:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-54} \lor \neg \left(y.im \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3700000000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-52} \lor \neg \left(y.im \leq 6.2 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.7e+147)
   (/ x.im y.im)
   (if (<= y.im -3700000000000.0)
     (/ (* y.re (/ x.re y.im)) y.im)
     (if (or (<= y.im -4.5e-52) (not (<= y.im 6.2e-15)))
       (/ x.im y.im)
       (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.7e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -3700000000000.0) {
		tmp = (y_46_re * (x_46_re / y_46_im)) / y_46_im;
	} else if ((y_46_im <= -4.5e-52) || !(y_46_im <= 6.2e-15)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.7d+147)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-3700000000000.0d0)) then
        tmp = (y_46re * (x_46re / y_46im)) / y_46im
    else if ((y_46im <= (-4.5d-52)) .or. (.not. (y_46im <= 6.2d-15))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.7e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -3700000000000.0) {
		tmp = (y_46_re * (x_46_re / y_46_im)) / y_46_im;
	} else if ((y_46_im <= -4.5e-52) || !(y_46_im <= 6.2e-15)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.7e+147:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -3700000000000.0:
		tmp = (y_46_re * (x_46_re / y_46_im)) / y_46_im
	elif (y_46_im <= -4.5e-52) or not (y_46_im <= 6.2e-15):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.7e+147)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -3700000000000.0)
		tmp = Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im);
	elseif ((y_46_im <= -4.5e-52) || !(y_46_im <= 6.2e-15))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.7e+147)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -3700000000000.0)
		tmp = (y_46_re * (x_46_re / y_46_im)) / y_46_im;
	elseif ((y_46_im <= -4.5e-52) || ~((y_46_im <= 6.2e-15)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.7e+147], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3700000000000.0], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[Or[LessEqual[y$46$im, -4.5e-52], N[Not[LessEqual[y$46$im, 6.2e-15]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.69999999999999998e147 or -3.7e12 < y.im < -4.5e-52 or 6.1999999999999998e-15 < y.im

   1. Initial program 48.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 71.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -2.69999999999999998e147 < y.im < -3.7e12

   1. Initial program 72.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 58.4%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   4. Taylor expanded in x.im around 0 41.2%

      \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im}}}{y.im} \]
   5. Step-by-step derivation

      1. *-commutative 41.2%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im}}{y.im} \]

      2. associate-*r/ 47.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]

   6. Simplified 47.4%

      \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]

   if -4.5e-52 < y.im < 6.1999999999999998e-15

   1. Initial program 63.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 68.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3700000000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-52} \lor \neg \left(y.im \leq 6.2 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -550000000000:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -8.6 \cdot 10^{-54} \lor \neg \left(y.im \leq 3.4 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.7e+147)
   (/ x.im y.im)
   (if (<= y.im -550000000000.0)
     (/ (/ y.re (/ y.im x.re)) y.im)
     (if (or (<= y.im -8.6e-54) (not (<= y.im 3.4e-11)))
       (/ x.im y.im)
       (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.7e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -550000000000.0) {
		tmp = (y_46_re / (y_46_im / x_46_re)) / y_46_im;
	} else if ((y_46_im <= -8.6e-54) || !(y_46_im <= 3.4e-11)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-5.7d+147)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-550000000000.0d0)) then
        tmp = (y_46re / (y_46im / x_46re)) / y_46im
    else if ((y_46im <= (-8.6d-54)) .or. (.not. (y_46im <= 3.4d-11))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.7e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -550000000000.0) {
		tmp = (y_46_re / (y_46_im / x_46_re)) / y_46_im;
	} else if ((y_46_im <= -8.6e-54) || !(y_46_im <= 3.4e-11)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.7e+147:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -550000000000.0:
		tmp = (y_46_re / (y_46_im / x_46_re)) / y_46_im
	elif (y_46_im <= -8.6e-54) or not (y_46_im <= 3.4e-11):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.7e+147)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -550000000000.0)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im / x_46_re)) / y_46_im);
	elseif ((y_46_im <= -8.6e-54) || !(y_46_im <= 3.4e-11))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.7e+147)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -550000000000.0)
		tmp = (y_46_re / (y_46_im / x_46_re)) / y_46_im;
	elseif ((y_46_im <= -8.6e-54) || ~((y_46_im <= 3.4e-11)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.7e+147], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -550000000000.0], N[(N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[Or[LessEqual[y$46$im, -8.6e-54], N[Not[LessEqual[y$46$im, 3.4e-11]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.69999999999999991e147 or -5.5e11 < y.im < -8.5999999999999999e-54 or 3.3999999999999999e-11 < y.im

   1. Initial program 48.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 71.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -5.69999999999999991e147 < y.im < -5.5e11

   1. Initial program 72.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 58.4%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   4. Taylor expanded in x.im around 0 41.2%

      \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im}}}{y.im} \]
   5. Step-by-step derivation

      1. *-commutative 41.2%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im}}{y.im} \]

      2. associate-*r/ 47.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]

   6. Simplified 47.4%

      \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]
   7. Step-by-step derivation

      1. clear-num 47.4%

         \[\leadsto \frac{y.re \cdot \color{blue}{\frac{1}{\frac{y.im}{x.re}}}}{y.im} \]

      2. un-div-inv 47.5%

         \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.re}}}}{y.im} \]

   8. Applied egg-rr 47.5%

      \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.re}}}}{y.im} \]

   if -8.5999999999999999e-54 < y.im < 3.3999999999999999e-11

   1. Initial program 63.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 68.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -550000000000:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -8.6 \cdot 10^{-54} \lor \neg \left(y.im \leq 3.4 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-17} \lor \neg \left(y.im \leq 9.6 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.5e-17) (not (<= y.im 9.6e+17)))
   (/ (+ x.im (* y.re (* x.re (/ 1.0 y.im)))) y.im)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.5e-17) || !(y_46_im <= 9.6e+17)) {
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.5d-17)) .or. (.not. (y_46im <= 9.6d+17))) then
        tmp = (x_46im + (y_46re * (x_46re * (1.0d0 / y_46im)))) / y_46im
    else
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.5e-17) || !(y_46_im <= 9.6e+17)) {
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.5e-17) or not (y_46_im <= 9.6e+17):
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.5e-17) || !(y_46_im <= 9.6e+17))
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re * Float64(1.0 / y_46_im)))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.5e-17) || ~((y_46_im <= 9.6e+17)))
		tmp = (x_46_im + (y_46_re * (x_46_re * (1.0 / y_46_im)))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.5e-17], N[Not[LessEqual[y$46$im, 9.6e+17]], $MachinePrecision]], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.50000000000000001e-17 or 9.6e17 < y.im

   1. Initial program 51.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 76.9%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. div-inv 76.8%

         \[\leadsto \frac{x.im + \color{blue}{\left(x.re \cdot y.re\right) \cdot \frac{1}{y.im}}}{y.im} \]

      2. *-commutative 76.8%

         \[\leadsto \frac{x.im + \color{blue}{\left(y.re \cdot x.re\right)} \cdot \frac{1}{y.im}}{y.im} \]

      3. associate-*l* 81.8%

         \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   5. Applied egg-rr 81.8%

      \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}}{y.im} \]

   if -5.50000000000000001e-17 < y.im < 9.6e17

   1. Initial program 65.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 82.9%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. associate-/l* 83.2%

         \[\leadsto \frac{x.re + \color{blue}{x.im \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-17} \lor \neg \left(y.im \leq 9.6 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{x.im + y.re \cdot \left(x.re \cdot \frac{1}{y.im}\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.6 \cdot 10^{-59} \lor \neg \left(y.im \leq 2.45 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.6e-59) (not (<= y.im 2.45e-15)))
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.6e-59) || !(y_46_im <= 2.45e-15)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-9.6d-59)) .or. (.not. (y_46im <= 2.45d-15))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.6e-59) || !(y_46_im <= 2.45e-15)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -9.6e-59) or not (y_46_im <= 2.45e-15):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9.6e-59) || !(y_46_im <= 2.45e-15))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -9.6e-59) || ~((y_46_im <= 2.45e-15)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9.6e-59], N[Not[LessEqual[y$46$im, 2.45e-15]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.6000000000000006e-59 or 2.45e-15 < y.im

   1. Initial program 54.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 74.5%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. associate-/l* 78.3%

         \[\leadsto \frac{x.im + \color{blue}{x.re \cdot \frac{y.re}{y.im}}}{y.im} \]

   5. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}} \]

   if -9.6000000000000006e-59 < y.im < 2.45e-15

   1. Initial program 63.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 68.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.6 \cdot 10^{-59} \lor \neg \left(y.im \leq 2.45 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{-17} \lor \neg \left(y.im \leq 1.08 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.4e-17) (not (<= y.im 1.08e+18)))
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.4e-17) || !(y_46_im <= 1.08e+18)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.4d-17)) .or. (.not. (y_46im <= 1.08d+18))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.4e-17) || !(y_46_im <= 1.08e+18)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.4e-17) or not (y_46_im <= 1.08e+18):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.4e-17) || !(y_46_im <= 1.08e+18))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.4e-17) || ~((y_46_im <= 1.08e+18)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.4e-17], N[Not[LessEqual[y$46$im, 1.08e+18]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.4000000000000002e-17 or 1.08e18 < y.im

   1. Initial program 51.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around inf 76.9%

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   4. Step-by-step derivation

      1. associate-/l* 81.0%

         \[\leadsto \frac{x.im + \color{blue}{x.re \cdot \frac{y.re}{y.im}}}{y.im} \]

   5. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}} \]

   if -5.4000000000000002e-17 < y.im < 1.08e18

   1. Initial program 65.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 82.9%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. associate-/l* 83.2%

         \[\leadsto \frac{x.re + \color{blue}{x.im \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{-17} \lor \neg \left(y.im \leq 1.08 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-54} \lor \neg \left(y.im \leq 1.75 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.2e-54) (not (<= y.im 1.75e-12)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.2e-54) || !(y_46_im <= 1.75e-12)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.2d-54)) .or. (.not. (y_46im <= 1.75d-12))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.2e-54) || !(y_46_im <= 1.75e-12)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.2e-54) or not (y_46_im <= 1.75e-12):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.2e-54) || !(y_46_im <= 1.75e-12))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.2e-54) || ~((y_46_im <= 1.75e-12)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.2e-54], N[Not[LessEqual[y$46$im, 1.75e-12]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.20000000000000008e-54 or 1.75e-12 < y.im

   1. Initial program 54.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 62.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -6.20000000000000008e-54 < y.im < 1.75e-12

   1. Initial program 63.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 68.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-54} \lor \neg \left(y.im \leq 1.75 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.im y.im))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_46im
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im / y_46_im

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_46_im / y_46_im)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im / y_46_im;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(x$46$im / y$46$im), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
2. Add Preprocessing

3. Taylor expanded in y.re around 0 38.6%

   \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

4. Final simplification 38.6%

   \[\leadsto \frac{x.im}{y.im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))